Suppose X is the waiting time for a failure of a component and X>0. The probability density function of X is f(x) and the cumulative distribution function of X is F(x). The hazard rate, or hazard function, of X is defined as h(x) = f(x)/[1-F(x)], where h(x)dx represents the conditional probability of X being in the interval (x, x+dx) given that the component has not failed up to time x. Suppose X has a Weibull probability density function where β-1 Bx x f(x) x2 0; B,0>0
= 0 elsewhere
a) Find the hazard rate of X
b) Suppose it is known that B 2.25. Further suppose a random sample of n values of X is available. Find the maximum likelihood estimator of the hazard rate, for a given value of x.